
The results obtained by Diethelem and Ford [1]} can be regarded as the particular cases of Theorem REF  and Theorem REF  and can be derived by putting \(\nu =1,~a=0 \)  and \(\Psi (t)=t, ~t\in \Lambda \) .

We have investigated in Theorem REF  and Theorem REF , the existence and uniqueness of the Cauchy problem (REF )-() involving the \(\Psi \) -Hilfer fractional derivative. One of the fundamental properties of the \(\Psi \) -Hilfer fractional derivative is the wide class of fractional derivatives that contain it as particular cases. Thus, in this sense, the respective results obtained by Theorem REF  and Theorem REF , are also valid for these fractional derivatives. For examples, some particular cases in the items as follows:

If we take \(\nu \rightarrow 1\)  on both sides of (REF )-(),it reduces to the Cauchy problem with \(\Psi \) -Caputo fractional derivative and consequently, the Theorem REF  and Theorem REF , there are true;

If we take \(\nu \rightarrow 0\)  on both sides of the (REF )-(), we have the Cauchy problem with \(\Psi \) -Riemann-Liouville fractional derivative and consequently, the Theorem REF  and Theorem REF , there are true;

Choose \(\Psi (t)=t^{\rho }\)  and take \(\nu \rightarrow 1\)  on both sides of (REF )-(), we have the Cauchy problem involving the Caputo-Katugampola fractional derivative and consequently, the Theorem REF  and Theorem REF , there are true;

In cases where they involve the fractional derivatives of Hadamard, Caputo-Hadamard, Hilfer-Hadamard or any other that is related to Hadamard's fractional derivative, we have to impose the condition on the parameter \(a>0\) , since in these fractional derivatives they involve the function \(\ln t\)  and is not defined when \(t=a=0\) .



